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sav08:qe_for_presburger_arithmetic [2009/04/23 09:44] vkuncak |
sav08:qe_for_presburger_arithmetic [2009/04/23 14:51] vkuncak |
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| We obtain a disjunction of conjunctions of literals of the form $0 < t$ and $K \mid t$ where $t$ are of the form $K_0 + \sum_{i=1}^n K_i \cdot x_i$ | We obtain a disjunction of conjunctions of literals of the form $0 < t$ and $K \mid t$ where $t$ are of the form $K_0 + \sum_{i=1}^n K_i \cdot x_i$ | ||
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| We first drop all constraints except divisibility, obtaining $F_2(x)$ | We first drop all constraints except divisibility, obtaining $F_2(x)$ | ||
| \[ | \[ | ||
| - | \bigwedge_{i=1}^D K_i \mid (x_i + t_i) | + | \bigwedge_{i=1}^D K_i \mid (x + t_i) |
| \] | \] | ||
| and then eliminate quantifier as | and then eliminate quantifier as | ||
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| $\exists res, i. \neg true$\\ | $\exists res, i. \neg true$\\ | ||
| $false$ | $false$ | ||
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| + | |||
| ===== Some Improvements ===== | ===== Some Improvements ===== | ||
| Avoid transforming to conjunctions of literals: work directly on negation-normal form. | Avoid transforming to conjunctions of literals: work directly on negation-normal form. | ||
| + | * the technique is similar to what we described for conjunctive normal form | ||
| - | See Section 7.2 of the [[Calculus of Computation Textbook]]. | + | This is the Cooper's algorithm |
| + | * {{sav09:reddyloveland78presburgerboundedalternation.pdf|Reddy, Loveland: Presburger Arithmetic with Bounded Quantifier Alternation}} (gives a slight improvement of the original Cooper's algorithm) | ||
| + | * Section 7.2 of the [[Calculus of Computation Textbook]] | ||
| ===== References ===== | ===== References ===== | ||